# Julia set

Let $U$ be an open subset of the complex plane and let $f\colon U\to U$ be analytic. Denote the $n$-th iterate of $f$ by $f^{n}$, i.e. $f^{1}=f$ and $f^{n+1}=f\circ f^{n}$. Then the Julia set of $f$ is the subset $J$ of $U$ characterized by the following property: if $z\in J$ then the restriction of $\{f^{n}\mid n\in\mathbb{N}\}$ to any neighborhood  of $z$ is not a normal family.

It can also be shown that the Julia set of $f$ is the closure of the set of repelling periodic points of $f$. (Repelling periodic point means that, for some $n$, we have $f^{n}(z)=z$ and $|f^{\prime}(z)|>1$.)

A simple example is afforded by the map $f(z)=z^{2}$; in this case, the Julia set is the unit circle. In general, however, things are much more complicated and the Julia set is a fractal.

From the definition, it follows that the Julia set is closed under $f$ and its inverse$f(J)=J$ and $f^{-1}(J)=J$. Topologically, Julia sets are perfect and have empty interior.

Title Julia set JuliaSet 2013-03-22 17:15:26 2013-03-22 17:15:26 rspuzio (6075) rspuzio (6075) 7 rspuzio (6075) Definition msc 28A80